Cone-shape distribution function

The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,[1] (acronymized as the ZAM [2][3][4] distribution[5] or ZAMD[1]), is one of the members of Cohen's class distribution function.[1][6] It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990.[7] The distribution's name stems from the twin cone shape of the distribution's kernel function on the ${\displaystyle t,\tau }$ plane.[8] The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel.[9][10]

Mathematical definition

The definition of the cone-shape distribution function is:

${\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,}$

where

${\displaystyle A_{x}(\eta ,\tau )=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi t\eta }\,dt,}$

and the kernel function is

${\displaystyle \Phi \left(\eta ,\tau \right)={\frac {\sin \left(\pi \eta \tau \right)}{\pi \eta \tau }}\exp \left(-2\pi \alpha \tau ^{2}\right).}$

The kernel function in ${\displaystyle t,\tau }$ domain is defined as:

${\displaystyle \phi \left(t,\tau \right)={\begin{cases}{\frac {1}{\tau }}\exp \left(-2\pi \alpha \tau ^{2}\right),&|\tau |\geq 2|t|,\\0,&{\mbox{otherwise}}.\end{cases}}}$

Following are the magnitude distribution of the kernel function in ${\displaystyle t,\tau }$ domain.

Following are the magnitude distribution of the kernel function in ${\displaystyle \eta ,\tau }$ domain with different ${\displaystyle \alpha }$ values.

As is seen in the figure above, a properly chosen kernel of cone-shape distribution function can filter out the interference on the ${\displaystyle \tau }$ axis in the ${\displaystyle \eta ,\tau }$ domain, or the ambiguity domain. Therefore, unlike the Choi-Williams distribution function, the cone-shape distribution function can effectively reduce the cross-term results form two component with same center frequency. However, the cross-terms on the ${\displaystyle \eta }$ axis are still preserved.

The cone-shape distribution function is in the MATLAB Time-Frequency Toolbox[11] and National Instruments' LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis [12]

See also

• Cohen's class distribution function
• Choi-Williams distribution function
• Wigner distribution function
• Ambiguity function
• Short-time Fourier transform

References

1. Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994)
2. L.M. Khadra; J. A. Draidi; M. A. Khasawneh; M. M. Ibrahim. (1998). "Time-frequency distributions based on generalized cone-shaped kernels for the representation of nonstationary signals". Journal of the Franklin Institute. 335 (5): 915–928. doi:10.1016/s0016-0032(97)00023-9.
3. Deze Zeng; Xuan Zeng; G. Lu; B. Tang (2011). "Automatic modulation classification of radar signals using the generalised time-frequency representation of Zhao, Atlas and Marks". IET Radar, Sonar & Navigation. 5 (4): 507–516. doi:10.1049/iet-rsn.2010.0174.
4. James R. Bulgrin; Bernard J. Rubal; Theodore E. Posch; Joe M. Moody. "Comparison of binomial, ZAM and minimum cross-entropy time-frequency distributions of intracardiac heart sounds". Signals, Systems and Computers, 1994. 1994 Conference Record of the Twenty-Eighth Asilomar Conference on. 1: 383–387.
5. Christos,Skeberis, Zaharias D. Zaharis, Thomas D. Xenos, and Dimitrios Stratakis. (2014). "ZAM distribution analysis of radiowave ionospheric propagation interference measurements". Telecommunications and Multimedia (TEMU), 2014 International Conference on: 155–161.CS1 maint: multiple names: authors list (link)
6. Leon Cohen (1989). "Time-frequency distributions-a review". Proceedings of the IEEE. 77 (7): 941–981. CiteSeerX 10.1.1.1026.2853. doi:10.1109/5.30749.
7. Y. Zhao; L. E. Atlas; R. J. Marks II (July 1990). "The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals". IEEE Transactions on Acoustics, Speech, and Signal Processing. 38 (7): 1084–1091. CiteSeerX 10.1.1.682.8170. doi:10.1109/29.57537.
8. R.J. Marks II (2009). Handbook of Fourier analysis & its applications. Oxford University Press.
9. Patrick J. Loughlin; James W. Pitton; Les E. Atlas (1993). "Bilinear time-frequency representations: New insights and properties". IEEE Transactions on Signal Processing. 41 (2): 750–767. Bibcode:1993ITSP...41..750L. doi:10.1109/78.193215.
10. Seho Oh; R. J. Marks II (1992). "Some properties of the generalized time frequency representation with cone-shaped kernel". IEEE Transactions on Signal Processing. 40 (7): 1735–1745. Bibcode:1992ITSP...40.1735O. doi:10.1109/78.143445.
11. Time-Frequency Toolbox For Use with MATLAB
12. National Instruments. LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis. TFA Cone-Shaped Distribution VI